An action principle for the Vlasov equation and associated Lie perturbation equations. Part 2. The Vlasov–Maxwell system

1993 ◽  
Vol 49 (2) ◽  
pp. 255-270 ◽  
Author(s):  
Jonas Larsson
Keyword(s):  
Distribution Function ◽  
Vlasov Equation ◽  
Maxwell Equations ◽  
Particle Distribution ◽  
Action Principle ◽  
Time Dependent ◽  
Maxwell System ◽  
Dynamical Equations ◽  
Hamiltonian Structures ◽  
Small Parameters

An action principle for the Vlasov–Maxwell system in Eulerian field variables is presented. Thus the (extended) particle distribution function appears as one of the fields to be freely varied in the action. The Hamiltonian structures of the Vlasov–Maxwell equations and of the reduced systems associated with small-ampliltude perturbation calculations are easily obtained. Previous results for the linearized Vlasov–Maxwell system are generalized. We find the Hermitian structure also when the background is time-dependent, and furthermore we may now also include the case of non-Hamiltonian perturbations within the Hamiltonian-Hermitian context. The action principle for the Vlasov–Maxwell system appears to be suitable for the derivation of reduced dynamical equations by expanding the action in various small parameters.

Kinetic theory

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Distribution Function ◽  
Kinetic Theory ◽  
Liouville Equation ◽  
Vlasov Equation ◽  
Particle Distribution ◽  
Equations Of Motion ◽  
Poisson Brackets ◽  
Hamiltonian Form ◽  
First Order ◽  
Number Of Particles

This chapter covers the equations governing the evolution of particle distribution and relates the macroscopic thermodynamical quantities to the distribution function. The motion of N particles is governed by 6N equations of motion of first order in time, written in either Hamiltonian form or in terms of Poisson brackets. Thus, as this chapter shows, as the number of particles grows it becomes necessary to resort to a statistical description. The chapter first introduces the Liouville equation, which states the conservation of the probability density, before turning to the Boltzmann–Vlasov equation. Finally, it discusses the Jeans equations, which are the equations obtained by taking various averages over velocities.

Kinetic equations for plasmas subjected to a strong time-dependent external field: Part 3. Stochastic equations for plasma turbulence

1975 ◽  
Vol 13 (1) ◽  
pp. 33-51 ◽  
Author(s):  
R. Balescu ◽  
J. H. Misguich
Keyword(s):  
Distribution Function ◽  
Kinetic Equation ◽  
External Field ◽  
Vlasov Equation ◽  
General Equation ◽  
Kinetic Equations ◽  
Plasma Turbulence ◽  
Stochastic Equations ◽  
Time Dependent ◽  
Average Distribution

The kinetic equation obtained in parts 1 and 2 is treated stochastically: the external field is stochastic, with an average and a fluctuating part. The turbulence of the system is described by the induced fluctuations in the plasma, and a general equation is derived for the average distribution function. As a particular case, the stochastic Vlasov equation is treated explicitly, and compared with the descriptions of Dupree, Weinstock and Benford & Thomson.

An action principle for the Vlasov equation and associated Lie perturbation equations. Part 1. The Vlasov—Poisson system

1992 ◽  
Vol 48 (1) ◽  
pp. 13-35 ◽  
Author(s):  
Jonas Larsson
Keyword(s):  
Vlasov Equation ◽  
Small Amplitude ◽  
Particle Distribution ◽  
Wave Interaction ◽  
Action Principle ◽  
Poisson System ◽  
Perturbation Calculation ◽  
Lagrange Equations ◽  
Theoretical Methods ◽  
Starting Point

A new action principle determining the dynamics of the Vlasov–Poisson system is presented (the Vlasov–Maxwell system will be considered in Part 2). The particle distribution function is explicitly a field to be varied in the action principle, in which only fundamentally Eulerian variables and fields appear. The Euler–Lagrange equations contain not only the Vlasov–Poisson system but also equations associated with a Lie perturbation calculation on the Vlasov equation. These equations greatly simplify the extensive algebra in the small-amplitude expansion. As an example, a general, manifestly Manley–Rowesymmetric, expression for resonant three-wave interaction is derived. The new action principle seems ideally suited for the derivation of action principles for reduced dynamics by the use of various averaging transformations (such as guiding-centre, oscillation-centre or gyro-centre transformations). It is also a powerful starting point for the application of field-theoretical methods. For example, the recently found Hermitian structure of the linearized equations is given a very simple and instructive derivation, and so is the well-known Hamiltonian bracket structure of the Vlasov–Poisson system.

scholarly journals Exact Statistical Mechanics of a One-Dimensional Self-Gravitating System

1971 ◽  
Vol 10 ◽  
pp. 56-72
Author(s):  
George B. Rybicki
Keyword(s):  
Distribution Function ◽  
Statistical Mechanics ◽  
Vlasov Equation ◽  
Total Mass ◽  
Particle Distribution ◽  
Spatial Density ◽  
One Dimensional ◽  
Large Numbers ◽  
Order Of Magnitude ◽  
The One

AbstractThe statistical mechanics of an isolated self-gravitating system consisting of N uniform mass sheets is considered using both canonical and microcanonical ensembles. The one-particle distribution function is found in closed form. The limit for large numbers of sheets with fixed total mass and energy is taken and is shown to yield the isothermal solution of the Vlasov equation. The order of magnitude of the approach to Vlasov theory is found to be 0(1/N). Numerical results for spatial density and velocity distributions are given.

Kinetic equations for plasmas subjected to a strong time-dependent external field. Part 1. General theory

1974 ◽  
Vol 11 (3) ◽  
pp. 357-375 ◽  
Author(s):  
R. Balescu ◽  
J. H. Misguich
Keyword(s):  
Distribution Function ◽  
Kinetic Equation ◽  
General Theory ◽  
External Field ◽  
Projection Operator ◽  
Nonlinear Response ◽  
Particle Distribution ◽  
Kinetic Equations ◽  
Time Dependent ◽  
Distribution Vector

It is shown that the concept of subdynamics introduced by Prigogine, George & Henin, and extended by Balescu & Wallenborn, can be generalized nontrivially to systems submitted to time-dependent external fields. The distribution vector of the system is split into two components by means of a time- dependent projection operator. Each of these obeys an independent equation of evolution. The description of the evolution of one of these components (the superkinetic component) can be reduced to a kinetic equation for a one-particle distribution function. It is shown that, when the external field vanishes for all times t ≤ t0, and if the system has reached a (field-free) equilibrium (or a ‘kinetic state’) at time t0, then for t ≥ t0 the kinetic equation derived here provides an exact and complete description of the evolution. A general expression for the nonlinear response of the system to the external field is derived.

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